Prove that `a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.`

Prove that |b+c a a b c+a b cc a+b|=4a b c

Prove: |b+c a a b c+a b cc a+b|=4a b c

If |1 1 1a b c a^3b^2c^3|=(a-b)(b-c)(c-a)(a+b+c),w h e r ea ,b ,c are different, then the determinant |1 1 1(x-a)^2(x-b)^2(x-c)^2(x-b)(x-c)(x-c)(x-a)(x-a)(x-b)| vanishes when a.a+b+c=0 b. x=1/3(a+b+c) c. x=1/2(a+b+c) d. x=a+b+c

If f(x)=a x^2+b x+a^2+b^2+c^2-a b-b c-c a ,w h e r ea ,b ,c are distinct reals, has imaginary roots then (a) 2(a-b)+(a-b)^2+(b-c)^2+(c-a)^2>0 (b) f(0)>0 (c) f(-1)>0 (d)f(0)<0

Suppose A, B, C are defined as A=a^2b+a b^2-a^2c-a c^2, B=b^2c+b c^2-a^2b-a b^2, a n dC=a^2c+a c^2-b^2c-b c^2, w h e r ea > b > c >0 and the equation A x^2+B x+C=0 has equal roots, then a ,b ,c are in AdotPdot b. GdotPdot c. HdotPdot d. AdotGdotPdot

Prove that a cos A+b cos B+c cos C=4R sin A sin B sin C

Identify the correct statement If a ,b ,c are odd integers (a) a+b+c cannot be zero (b) If a ,b ,c are odd integers a^2+b^2-c^2!=0 (c) If a^2+b^2-c^2 , then the least one of a ,b ,c is even, given that a ,b ,c are integers (d) If a^2+b^2=c^2w h e r ea ,b ,c are integers then c > a+b

If a , b , c are real numbers, prove that |a b c b c a c a b|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w), where w is a complex cube root of unity.

Prove: |(a^2,b c, a c+c^2),(a^2+a b,b^2,a c ),(a b,b^2+b c,c^2)|=4a^2b^2c^2

In isosceles triangles A B C ,| vec A B|=| vec B C|=8, a point E divides A B internally in the ratio 1:3, then find the angle between vec C Ea n d vec C A(w h e r e| vec C A|=12)dot